Nuprl Lemma : constant-limit
∀a,b:ℝ. (lim n→∞.a = b ⇐⇒ a = b)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
sq_stable: SqStable(P),
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
nat: ℕ,
sq_exists: ∃x:{A| B[x]},
uimplies: b supposing a,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
converges-to_wf,
nat_wf,
nat_plus_wf,
req_wf,
real_wf,
false_wf,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
absval_wf,
rinv_wf2,
rmul_wf,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
uiff_transitivity2,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
real_term_value_mul_lemma,
rinv-as-rdiv,
squash_wf,
true_wf,
rabs-int,
infinitesmal-difference,
sq_stable__all,
sq_stable__rleq,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
imageElimination,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
applyEquality,
independent_pairEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
independent_functionElimination,
because_Cache,
inrFormation,
natural_numberEquality,
functionEquality,
rename,
setElimination,
dependent_functionElimination,
isect_memberFormation,
independent_isectElimination,
productElimination,
lemma_by_obid,
dependent_set_memberFormation,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}a,b:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.a = b \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2017_10_03-AM-09_05_04
Last ObjectModification:
2017_07_28-AM-07_41_19
Theory : reals
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